Re: Pauls fingerboard kit

> > Anyone want a 46 PB? This would be a good way to get one.
>
to which Pauul said...
>
> Awesome! I'm thinking that 22, 31,
> 46, and 72 would be a good choice
> of four cardinalities to base my
> future instruments on. My
> present plan for the four
> fingerboards that come with the
> Rankin system is 22-tET, 31-tET,
> 22-out-of-46-tET, and 31-out-of-
> 72-tET.
>

Hi Paul,

Could you post the snail or email address for the Rankin
system again? This is sounding very interesting...

Regarding 46 which just showed up all over this list...
I got interested in 41 over the weekend, and decided that
when I got tired of meantone (31), that will be my trip
to the 9-limit.

What does 46 have that everyone seems gaga about? Is 46
better at 13 (recognizing that it doesn't do 11 or 13
uniquely, it does keep them off of each other, which
41 does not).

Is your 22-out-of-46 a sort of "nontempered" version of
the PB that you see in 22tet?

Is the 31-of-72 going to be Canasta?

Bob Valentine

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:

> What does 46 have that everyone seems gaga about?
>
> Is your 22-out-of-46 a sort of "nontempered" version of
> the PB that you see in 22tet?

Sort of . . . 22-out-of-46 is a "srutar", for Indian-type music. The
diaschisma 2048:2025 must vanish. It doesn't in 41.

> Is the 31-of-72 going to be Canasta?

Yes, with split frets, so that standard tuning can be used for the
open strings.

Bob Valentine wrote:

> What does 46 have that everyone seems gaga about? Is 46
> better at 13 (recognizing that it doesn't do 11 or 13
> uniquely, it does keep them off of each other, which
> 41 does not).

46 is almost 15-limit consistent, but tends to be overshadowed by its
neighbours 41 and 58. Well, 58 isn't casting much of a shadow right now,
but that's all going to change. So, every now and then we have to remind
ourselves that 46 is quite consonant as well, although those of us who
care about that are probably going back to 41 or 58 or 31 or 72.

Interestingly, fudging my temperament finding script so that 46 gets
included with the 15-limit ETs does bring 29+46 well into the rankings. I
wonder what else we could be missing by insisting on consistency.
Probably no killer temperaments, but maybe some keyboard mappings. For
example, a meantone-31 can do the 11-limit with 19 notes, but it doesn't
make the list because 31 is the only consistent temperament that works
with it. Also it isn't unique, but you can't have everything.

Graham

Graham wrote...

>46 is almost 15-limit consistent, but tends to be overshadowed by
>its neighbours 41 and 58. Well, 58 isn't casting much of a shadow
>right now, but that's all going to change.

What's going to change it?

-C.

Carl wrote:

> Graham wrote...
>
> >46 is almost 15-limit consistent, but tends to be overshadowed by
> >its neighbours 41 and 58. Well, 58 isn't casting much of a shadow
> >right now, but that's all going to change.
>
> What's going to change it?

Oh, it's bound to change. You can't keep a temperament like that down for
long. Wait until I get my hands on a ZTar ...

Graham

--- In tuning-math@y..., graham@m... wrote:
> Bob Valentine wrote:
>
> > What does 46 have that everyone seems gaga about? Is 46
> > better at 13

Yes, but I'm not planning on using it for 13-limit. More important is
that 46 is better than 41 in 5-limit (which is the main interval
flavor for Indian music). Though 34 is better still in the 5-limit,
46 allows me to access 7- and 11-flavors with much higher accuracy.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > Bob Valentine wrote:

> > > What does 46 have that everyone seems gaga about? Is 46
> > > better at 13

> Yes, but I'm not planning on using it for 13-limit. More important
is
> that 46 is better than 41 in 5-limit (which is the main interval
> flavor for Indian music). Though 34 is better still in the 5-limit,
> 46 allows me to access 7- and 11-flavors with much higher accuracy.

Here are relativized n-consistent goodness measures for odd n to 25,
for 41, 46 and 58:

41:

3 .67803586
5 1.380445520
7 .7433989824
9 .8004237371
11 .9169187332
13 1.010934526
15 1.010934526
17 1.138442704
19 1.042105199
21 1.042105199
23 1.399948567
25 1.399948567

46:

3 4.21934770
5 1.297511950
7 1.181094581
9 1.181094581
11 .8584628865
13 .8850299838
15 1.082290028
17 .9526171496
19 1.188323665
21 1.188323665
23 1.109794688
25 1.270499309

58:

3 4.18614652
5 2.499272068
7 1.270307523
9 1.270307523
11 .9740696116
13 .8435935409
15 .9018214272
17 .9309801788
19 1.393450457
21 1.393450457
23 1.295990287
25 1.721256452

We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits;
46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an
excellent system for 7, 9 or 11, and deserves some respect!

In-Reply-To: <9pbgq1+86qa@eGroups.com>
Gene wrote:

> Here are relativized n-consistent goodness measures for odd n to 25,
> for 41, 46 and 58:

How are these calculated? It looks like lower numbers are better, so it's
really a badness measure.

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9pbgq1+86qa@e...>
> Gene wrote:

> > Here are relativized n-consistent goodness measures for odd n to
25,
> > for 41, 46 and 58:

> How are these calculated? It looks like lower numbers are better,
so it's
> really a badness measure.

This is the same w-consistent measure cons(w, n) introduced in
/tuning-math/message/860
And yes, it is a badness measure, but you could always take the
reciprocal. :)

In-Reply-To: <9pcpdn+pinr@eGroups.com>
> Gene wrote:

> This is the same w-consistent measure cons(w, n) introduced in
> /tuning-math/message/860
> And yes, it is a badness measure, but you could always take the
> reciprocal. :)

And "less than or equal to l" should be "less than or equal to w". I was
wondering how a prime could be less than 1.

It looks like h(q_i) is the number of steps in the ET for the ratio q_i.
So h(2) is the number of steps to the octave. So cons(w, n) assumes
consistency which -- warning! -- doesn't hold for 46-equal in the
15-limit, because there's one ambiguous interval.

This formula:

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))

has a parenthesis unclosed. From the one lower down, I think it should be

n^(1/d) * max(abs(n*log_2(q_i)) - h(q_i))

The n^(1/d) means you're scaling according to the size of the octave and
the number of prime dimensions.

Then you're taking the largest deviation for any interval within the
limit, right? In which case, the parenthesising should be

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i)))

and the other formula must be wrong.

That could be re-written

n^(1/d) * n * max(|tempered_pitch - just_pitch|)

where pitches are in octaves. Which can be simplified to

n^(1+1/d) * max(|tempered_pitch - just_pitch|)

Is that right? I think I almost understand it! So why the 1+1/d?

Graham

--- In tuning-math@y..., genewardsmith@j... wrote:

> 41 is clearly an
> excellent system for 7, 9 or 11, and deserves some respect!

Oh yes . . . but it's mentioned far more than 46 in the literature,
which is why I suspect Robert Valentine was puzzled about the 46
hoopla.

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., graham@m... wrote:
> > > Bob Valentine wrote:
>
> > > > What does 46 have that everyone seems gaga about? Is 46
> > > > better at 13
>
> > Yes, but I'm not planning on using it for 13-limit. More
important
> is
> > that 46 is better than 41 in 5-limit (which is the main interval
> > flavor for Indian music). Though 34 is better still in the 5-
limit,
> > 46 allows me to access 7- and 11-flavors with much higher
accuracy.
>
> Here are relativized n-consistent goodness measures for odd n to
25,
> for 41, 46 and 58:
>
> 41:
>
> 3 .67803586
> 5 1.380445520
> 7 .7433989824
> 9 .8004237371
> 11 .9169187332
> 13 1.010934526
> 15 1.010934526
> 17 1.138442704
> 19 1.042105199
> 21 1.042105199
> 23 1.399948567
> 25 1.399948567
>
> 46:
>
> 3 4.21934770
> 5 1.297511950
> 7 1.181094581
> 9 1.181094581
> 11 .8584628865
> 13 .8850299838
> 15 1.082290028
> 17 .9526171496
> 19 1.188323665
> 21 1.188323665
> 23 1.109794688
> 25 1.270499309
>
> 58:
>
> 3 4.18614652
> 5 2.499272068
> 7 1.270307523
> 9 1.270307523
> 11 .9740696116
> 13 .8435935409
> 15 .9018214272
> 17 .9309801788
> 19 1.393450457
> 21 1.393450457
> 23 1.295990287
> 25 1.721256452
>
> We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits;
> 46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an
> excellent system for 7, 9 or 11, and deserves some respect!

Gene, I don't know what you mean by n-consistent here. 46 is only
consistent through the 13-limit, so it's inconsistent in the 17-
limit. So how can 46 have a finite value for "17-consistent goodness"?

--- In tuning-math@y..., graham@m... wrote:

> And "less than or equal to l" should be "less than or equal to w".
I was
> wondering how a prime could be less than 1.

No, I mean that for those values of w, cons(w, n) is less than 1.

> It looks like h(q_i) is the number of steps in the ET for the ratio
q_i.
> So h(2) is the number of steps to the octave. So cons(w, n)
assumes
> consistency which -- warning! -- doesn't hold for 46-equal in the
> 15-limit, because there's one ambiguous interval.

Consistency always holds automatically the way I define things, since
I calculate everything based on the generators--that is, "h" is
defined not just by h(2) but also h(3), h(5) etc.

> The n^(1/d) means you're scaling according to the size of the
octave and
> the number of prime dimensions.

Right.

> That could be re-written
>
> n^(1/d) * n * max(|tempered_pitch - just_pitch|)
>
> where pitches are in octaves. Which can be simplified to

Right again.

> n^(1+1/d) * max(|tempered_pitch - just_pitch|)
>
> Is that right? I think I almost understand it! So why the 1+1/d?

If you are doing a simultaneous Diophantine approximation of d
different numbers, at least one of them irrational, by numbers with
denominator n, then we can find an infinite number of n such that all
of the numbers are within 1/n^(1+1/d) of the correct value. Choosing
a p-limit scale division involves the simultaneous approximation of d
values log_2(3), ..., log_2(p) by numbers of the form m/n. In this
case we add the condition that we also want to approximate
log_2(5/3), etc. but to get the dimensions right I still scale by
n^(1+1/d), since only d of these are independent.